Independent solution

How to solve this Conditional Probability question

Setup

Setup

Among the 20 questions, the student answers the known questions correctly and guesses correctly on half of the remaining questions.

Pr⁡(K∣C)=NN+12(20−N)=0.824\Pr(K\mid C)=\frac{N}{N+\frac12(20-N)}=0.824

Model

Model

Conditional on a correct response, the probability that it came from the known set is the expected number of known correct responses divided by the expected total number of correct responses.

N10+12N≈0.824\frac{N}{10+\frac12N}\approx0.824

Compute

Compute

Solving the resulting equation gives 14. Substitution gives fourteen seventeenths, approximately 0.824, confirming the required rounding.

N=14,1414+3=0.823529≈0.824N=14,\quad \frac{14}{14+3}=0.823529\approx0.824

Answer

Answer

The student knows 14 of the answers.

14(C)\boxed{14\quad\text{(C)}}